Laplace Transform of Dirac Delta Function/Warning
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Warning on Laplace Transform of Dirac Delta Function
Mathematically speaking, $\ds \lim_{\epsilon \mathop \to 0} \map {F_\epsilon} t$ does not actually exist.
Hence $\ds \laptrans {\lim_{\epsilon \mathop \to 0} \map {F_\epsilon} t}$ is not actually defined.
However, it is useful to consider $\map \delta t = \ds \lim_{\epsilon \mathop \to 0} \map {F_\epsilon} t$ to be such that $\laptrans {\map \delta t} = 1$.
Sources
- 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Solved Problems: Impulse Functions. The Dirac Delta Function: $42 \ \text{(b)}$